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arXiv · 2507.07038

Tilings of the sphere by congruent pentagons V: Edge combination $a^{4}b$ with rational angles

Jinjin Liang·Yixi Liao·Wenchuan Hu·Erxiao Wang
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Abstract

We classify edge-to-edge tilings of the sphere by congruent pentagons with the edge combination $a^4b$ and with rational angles in degree: they are a one-parameter family of symmetric $a^4b$-pentagonal subdivisions of the tetrahedron with $12$ tiles; a sequence of unique symmetric $a^4b$-pentagons admitting a symmetric $3$-layer earth map tiling by $4m$ tiles for any $m\ge4$, among which each odd $m$ case admits two standard flip modifications; and a unique non-symmetric and degenerate $a^4b$-pentagon admitting a non-symmetric $3$-layer earth map tiling and its standard flip modification with $20$ tiles. The full classification from this series and all induced non-edge-to-edge quadrilateral tilings from degenerate pentagons are summarized with their 3D pictures.

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Jinjin Liang, Yixi Liao, Wenchuan Hu, Erxiao Wang. 2025-07-09. Tilings of the sphere by congruent pentagons V: Edge combination $a^{4}b$ with rational angles. https://arxiv.org/abs/2507.07038

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